INO-07

Born–Landé lattice energy

U = −(NA α k |z+ z−| e² / r0)(1 − 1/n). Coulomb plus inverse-power repulsion.

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LatticeBorn–Landé

Governing equation

U=-\dfrac{N_A\alpha k|z_+z_-|e^2}{r_0}\left(1-\dfrac1n\right)

where

\alpha: Madelung α (—)
z_+: Cation charge (—)
z_-: Anion charge (—)
r_0: r0 (Å)
n: Born exponent (—)
U: Lattice energy (kJ/mol)

Lecture brief

Historical brief

Pauling and Allred–Rochow electronegativity, Kapustinskii lattice energy, CFSE and Goldschmidt radii organise the periodic solid. The sheets predict bond character and crystal packing. This sheet (INO-07 — Born–Landé lattice energy) is the form associated with Born–Landé. Working symbols: $\alpha$ , $z_+$ , $z_-$ , $r_0$ , $n$ $\rightarrow$ $U$ . The Born exponent n is ~5–12 from compressibility. n → ∞ recovers the pure Madelung well.

Purpose

Purpose: compute

U

from

\alpha

z_+

z_-

r_0

n

in Inorganic chemistry via

U=-\dfrac{N_A\alpha k|z_+z_-|e^2}{r_0}\left(1-\dfrac1n\right)

U = −(NA α k |z+ z−| e² / r0)(1 − 1/n). Coulomb plus inverse-power repulsion. Use it when a real inorganic chemistry question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

\alpha = 1.748\,\mathrm{—}

z_+ = 1.000\,\mathrm{—}

z_- = 1.000\,\mathrm{—}

r_0 = 2.820\,\mathrm{Å}

n = 9.000\,\mathrm{—}

, the governing relation

U=-\dfrac{N_A\alpha k|z_+z_-|e^2}{r_0}\left(1-\dfrac1n\right)

yields

U = -765.5\,\mathrm{kJ/mol}

. Binary crystal, r0 in Å, charges as integers. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Madelung α \alpha1.748 —Cation charge z_+1 —Anion charge z_-1 —r0 r_02.82 ÅBorn exponent n9 —

Outputs

Lattice energy U-765.5 kJ/mol

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Narration of this film

Binary crystal, r0 in Å, charges as integers.

The Born exponent n is ~5–12 from compressibility. n → ∞ recovers the pure Madelung well.

Reading speed

Watch on YouTube